This is a snapshot from the book "Fundamentals of Kalman Filtering: A Practical Approach Book by Paul Zarchan."
For a system: $\dot{x} = Fx + Qu + w$, the table 4.2 lists the discrete fundamental matrix $\Phi_k = \Phi (T_s)$ where $\Phi = e^{Ft}$, and $Q_k = \int_0^{T_s}\Phi(\tau) Q \Phi^T(\tau) d\tau$ where $T_s$ is the sampling time of the measurements.
Are there any quick methods to compute these terms for any given $F$? Assuming that $Q = \mathbf{0}$. I'm hoping there are resources that may already have these terms in a different form?
Yes there are techniques to do this. One method is the numerical technique developed by C.F Van Loan in this paper and further explained in Section 3.9 of this book. A brief description of the technique is as follows:
Suppose we are given a continuous time system $\dot{x} = Fx + Qu$, where $x$ is the $ n \times 1$ state vector and $u$ is the driving white noise vector with power spectral density matrix $W$.
1) First, compute the following $2n \times 2n$ matrix
$ A = \begin{bmatrix} -F & QWQ^T \\ 0 & F^T \end{bmatrix} \Delta t $
$\Delta t $ is the sampling time of the system.
2) Next compute the matrix exponential of A. MATLAB has a built-in function
expm(A)$M = e^A = \begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix} = \begin{bmatrix} \cdot \cdot \cdot & \Phi_k^{-1}Q_k \\ 0 & \Phi_k^T \end{bmatrix} $
3) According to this technique, the lower right partition of $M$ is the transpose of the state transition matrix. Thus,
$\Phi_k = B_{22}^T$
4) The discrete process noise matrix is obtained as:
$Q_k = \Phi_k B_{12}$
Hope this helps.